The Tsiolkovsky rocket equation, or ideal rocket equation, describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself (a thrust) by expelling part of its mass with high speed and move due to the conservation of momentum. The equation relates the delta-v (the maximum change of speed of the rocket if no other external forces act) with the effective exhaust velocity and the initial and final mass of a rocket (or other reaction engine).

For any such maneuver (or journey involving a number of such maneuvers):

Derivation

In the following derivation, "the rocket" is taken to mean "the rocket and all of its unburned propellant".

Newton's second law of motion relates external forces () to the change in linear momentum of the system as follows:

where is the momentum of the rocket at time t = 0:

and is the momentum of the rocket and exhausted mass at time :

and where, with respect to the observer:

is the velocity of the rocket at time t = 0

is the velocity of the rocket at time

is the velocity of the mass added to the exhaust (and lost by the rocket) during time

is the mass of the rocket at time t = 0

is the mass of the rocket at time

The velocity of the exhaust in the observer frame is related to the velocity of the exhaust in the rocket frame by (since exhaust velocity is in the negative direction)

Solving yields:

and, using , since ejecting a positive results in a decrease in mass,

If there are no external forces then and

Assuming is constant, this may be integrated to yield:

or equivalently

or or

where is the initial total mass including propellant, the final total mass, and the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration).

(delta v)
is the integration over time of the magnitude of the acceleration
produced by using the rocket engine (what would be the actual
acceleration if external forces were absent). In free space, for the
case of acceleration in the direction of the velocity, this is the
increase of the speed. In the case of an acceleration in opposite
direction (deceleration) it is the decrease of the speed.

Of course
gravity and drag also accelerate the vehicle, and they can add or
subtract to the change in velocity experienced by the vehicle. Hence
delta-v is not usually the actual change in speed or velocity of the
vehicle.

If special relativity is taken into account, the following equation can be derived for a relativistic rocket, with again standing for the rocket's final velocity (after burning off all its fuel and being reduced to a rest mass of ) in the inertial frame of reference where the rocket started at rest (with the rest mass including fuel being initially), and standing for the speed of light in a vacuum:

Writing as , a little algebra allows this equation to be rearranged as

Delta-v is of fundamental importance in orbital mechanics. It quantifies how difficult it is to perform a given orbital maneuver. To achieve a large delta-v, either must be huge (growing exponentially as delta-v rises), or must be tiny, or must be very high, or some combination of all of these.

In practice, very-high delta-v has been achieved by a combination of 1) very large rockets (increasing ), 2) staging (decreasing ), and 3) very high exhaust velocities.

The ion thruster
is an example of a high exhaust velocity rocket. Instead of storing
energy in the propellant itself as in a chemical rocket, ion and other
electric rockets separate energy storage from the reaction (propellant)
mass storage.

Not only does this allow very large (and in principle
unlimited) amounts of energy to be applied to small amounts of ejected
mass to achieve very high exhaust velocities, but energy sources far
more compact than chemical fuels can be used, such as nuclear reactors. In the inner solar system solar power can be used, entirely eliminating the need for a large internal primary energy storage system.

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